Integral equation approach for a hydrogen atom in a strong magnetic field

TL;DR

This paper develops an integral equation method to analyze a hydrogen atom in strong magnetic fields, capturing its anisotropic quantum behavior and providing numerical results across various field strengths.

Contribution

It introduces an integral equation approach that accurately models the anisotropic asymptotic behavior of a hydrogen atom in strong magnetic fields.

Findings

Exact analytic evaluation of Green's operators using continued fractions.

Calculation of the total Green's operator via complex contour integration.

Numerical results demonstrating the method's effectiveness across magnetic field strengths.

Abstract

The problem of a hydrogen atom in a strong magnetic field is a notorious example of a quantum system that has genuinely different asymptotic behaviors in different directions. In the direction perpendicular to the magnetic field the motion is quadratically confined, while in the direction along the field line the motion is a Coulomb-distorted free motion. In this work, we identify the asymptotically relevant parts of the Hamiltonian and cast the problem into a Lippmann-Schwinger form. Then, we approximate the asymptotically irrelevant parts by a discrete Hilbert space basis that allows an exact analytic evaluation of the relevant Green's operators by continued fractions. The total asymptotic Green's operator is calculated by a complex contour integral of subsystem Green's operators. We present a sample of numerical results for a wide range of magnetic field strengths.